A least-squares implicit RBF-FD closest point method and applications to PDEs on moving surfaces

The closest point method (Ruuth and Merriman, J. Comput. Phys. 227(3):1943-1961, [2008]) is an embedding method developed to solve a variety of partial differential equations (PDEs) on smooth surfaces, using a closest point representation of the surface and standard Cartesian grid methods in the embedding space. Recently, a closest point method with explicit time-stepping was proposed that uses finite differences derived from radial basis functions (RBF-FD). Here, we propose a least-squares implicit formulation of the closest point method to impose the constant-along-normal extension of the solution on the surface into the embedding space. Our proposed method is particularly flexible with respect to the choice of the computational grid in the embedding space. In particular, we may compute over a computational tube that contains problematic nodes. This fact enables us to combine the proposed method with the grid based particle method (Leung and Zhao, J. Comput. Phys. 228(8):2993-3024, [2009]) to obtain a numerical method for approximating PDEs on moving surfaces. We present a number of examples to illustrate the numerical convergence properties of our proposed method. Experiments for advection-diffusion equations and Cahn-Hilliard equations that are strongly coupled to the velocity of the surface are also presented

Molecular dynamics in arbitrary geometries : parallel evaluation of pair forces

A new algorithm for calculating intermolecular pair forces in molecular dynamics (MD) simulations on a distributed parallel computer is presented. The arbitrary interacting cells algorithm (AICA) is designed to operate on geometrical domains defined by an unstructured, arbitrary polyhedral mesh that has been spatially decomposed into irregular portions for parallelisation. It is intended for nano scale fluid mechanics simulation by MD in complex geometries, and to provide the MD component of a hybrid MD/continuum simulation. The spatial relationship of the cells of the mesh is calculated at the start of the simulation and only the molecules contained in cells that have part of their surface closer than the cut-off radius of the intermolecular pair potential are required to interact. AICA has been implemented in the open source C++ code OpenFOAM, and its accuracy has been indirectly verified against a published MD code. The same system simulated in serial and in parallel on 12 and 32 processors gives the same results. Performance tests show that there is an optimal number of cells in a mesh for maximum speed of calculating intermolecular forces, and that having a large number of empty cells in the mesh does not add a significant computational overhead

Perspectives on the simulation of micro gas and nano liquid flows

Micro- and nano-scale fluid systems can behave very differently from their macro-scale counterparts. Remarkably, there is no sufficiently accurate, computationally efficient, and — most importantly — generally agreed fluid dynamic model that encapsulates all of this important behaviour. The only thing that researchers can agree on is that the conventional Navier-Stokes fluid equations are unable to capture the unique complexity of these often locally non-thermodynamic-equilibrium flows. Here, we outline recent work on developing and exploring new models for these flows, highlighting, in particular, slip flow as a quintessential non-equilibrium (or sub-continuum) phenomenon. We describe the successes and failures of various hydrodynamic and molecular models in capturing the non-equilibrium flow physics in current test applications in micro and nano engineering, including the aerodynamic drag of a sphere in a rarefied gas, and the flow of water along carbon nanotubes

Hydrodynamics of Micro-swimmers in Films

One of the principal mechanisms by which surfaces and interfaces affect microbial life is by perturbing the hydrodynamic flows generated by swimming. By summing a recursive series of image systems we derive a numerically tractable approximation to the three-dimensional flow fields of a Stokeslet (point force) within a viscous film between a parallel no-slip surface and no-shear interface and, from this Green's function, we compute the flows produced by a force- and torque-free micro-swimmer. We also extend the exact solution of Liron & Mochon (1976) to the film geometry, which demonstrates that the image series gives a satisfactory approximation to the swimmer flow fields if the film is sufficiently thick compared to the swimmer size, and we derive the swimmer flows in the thin-film limit. Concentrating on the thick film case, we find that the dipole moment induces a bias towards swimmer accumulation at the no-slip wall rather than the water-air interface, but that higher-order multipole moments can oppose this. Based on the analytic predictions we propose an experimental method to find the multipole coefficient that induces circular swimming trajectories, allowing one to analytically determine the swimmer's three-dimensional position under a microscope.Comment: 35 pages, 11 figures, 5 table

Analytical ray-tracing in planetary atmospheres

Ground-based astro-geodetic observations and atmospheric occultations, are two examples of observational techniques requiring a scrutiny analysis of atmospheric refraction. In both cases, the measured changes in observables are geometrically related to changes in the photon path and the light time of the received electromagnetic signal. In the context of geometrical optics, the change in the physical properties of the signal are related to the refractive profile of the crossed medium. Therefore, having a clear knowledge of how the refractivity governs the photon path and the light time evolution is of prime importance to clearly understand observational features. Analytical studies usually focused on spherically symmetric atmospheres and only few aimed at exploring the effect of the non-spherical symmetry on the observables. In this paper, we analytically perform the integration of the photon path and the light time of rays traveling across a planetary atmosphere. We do not restrict our attention to spherically symmetric atmospheres and introduce a comprehensive mathematical framework which allows to handle any kind of analytical studies in the context of geometrical optics. To highlight the capabilities of this new formalism, we carry out five realistic applications for which we derive analytical solutions. The accuracy of the method of integration is assessed by comparing our results to a numerical integration of the equations of geometrical optics in the presence of a quadrupolar moment $J_2$. This shows that the analytical solution leads to the determination of the light time and the refractive bending with relative errors at the level of one part in $10^8$ and one part in $10^5$, for typical values of the refractivity and the $J_2$ parameter at levels of $10^{-4}$ and $10^{-2}$, respectively

Dynamics and Topological Aspects of a Reconstructed Two-Dimensional Foam Time Series Using Potts Model on a Pinned Lattice

We discuss a method to reconstruct an approximate two-dimensional foam structure from an incomplete image using the extended Potts mode with a pinned lattice we introduced in a previous paper. The initial information consists of the positions of the vertices only. We locate the centers of the bubbles using the Euclidean distance-map construction and assign at each vertex position a continuous pinning field with a potential falling off as $1/r$. We nucleate a bubble at each center using the extended Potts model and let the structure evolve under the constraint of scaled target areas until the bubbles contact each other. The target area constraint and pinning centers prevent further coarsening. We then turn the area constraint off and let the edges relax to a minimum energy configuration. The result is a reconstructed structure very close to the simulation. We repeated this procedure for various stages of the coarsening of the same simulated foam and investigated the simulation and reconstruction dynamics, topology and area distribution, finding that they agree to good accuracy.Comment: 31 pages, 20 Postscript figures Accepted in the Journal of Computational Physic

"Regularity Singularities" and the Scattering of Gravity Waves in Approximate Locally Inertial Frames

It is an open question whether solutions of the Einstein-Euler equations are smooth enough to admit locally inertial coordinates at points of shock wave interaction, or whether "regularity singularities" can exist at such points. The term {\it regularity singularity} was proposed by the authors as a point in spacetime where the gravitational metric tensor is Lipschitz continuous ($C^{0,1}$), but no smoother, in any coordinate system of the $C^{1,1}$ atlas. An existence theory for shock wave solutions in $C^{0,1}$ admitting arbitrary interactions has been proven for the Einstein-Euler equations in spherically symmetric spacetimes, but $C^{1,1}$ is the requisite smoothness required for space-time to be locally flat. Thus the open problem of regularity singularities is the problem as to whether locally inertial coordinate systems exist at shock waves within the larger $C^{1,1}$ atlas. To clarify this open problem, we identify new "Coriolis type" effects in the geometry of $C^{0,1}$ shock wave metrics and prove they are essential in the sense that they can never be made to vanish within the atlas of {\it smooth} coordinate transformations, the atlas usually assumed in classical differential geometry. Thus the problem of existence of regularity singularities is equivalent to the question as to whether or not these Coriolis type effects are essentially non-removable and `real', or merely coordinate effects that can be removed, (in analogy to classical Coriolis forces), by going to the less regular atlas of $C^{1,1}$ transformations. If essentially non-removable, it would argue strongly for a `real' new physical effect for General Relativity, providing a physical context to the open problem of regularity singularities.Comment: 29 pages. Version 2: Corrections of some typographical errors and improvements of wording. Results are unchange